Problem 50 - College Algebra

Question

In 2003 , the owl population in a park was measured to be 340. By 2007, the population was measured again to be \(285 .\) The population changes linearly. Let the input be years since 1990 . a. Find a formula for the owl population, \(P .\) Let the input be years since 2003 . b. What does your model predict the owl population to be in 2012?

Step-by-Step Solution

Answer

The owl population model is \( P(x) = -13.75x + 340 \). In 2012, the predicted population is 216.

Step 1: Identify Variables and Formulate Population Function

Let the input be years since 2003, represented as \( x \). The owl population \( P \) decreases linearly from 340 in 2003 to 285 in 2007, corresponding to 0 years and 4 years since 2003, respectively. We start by determining the slope \( m \) of the line, which is calculated as: \( m = \frac{285 - 340}{4-0} = -13.75 \). The linear equation can be written in the form \( P(x) = mx + b \), where \( b \) is the population in 2003, 340. This gives us \( P(x) = -13.75x + 340 \).

Step 2: Model Owl Population for Future Year

Using the function \( P(x) = -13.75x + 340 \) found previously, substitute \( x = 9 \) for the year 2012, which is 9 years since 2003. Calculate \( P(9) = -13.75 \times 9 + 340 = -123.75 + 340 = 216.25 \). Therefore, the model predicts the owl population to be approximately 216.

Key Concepts

Slope Calculation

The concept of slope is essential when dealing with linear equations and models. When working with population models, the slope helps us understand how quickly the population changes over time. To find the slope, we use the formula:

• slope \( m = \frac{\text{change in population}}{\text{change in years}} \)

In our exercise, the owl population decreased from 340 to 285 from 2003 to 2007. This span covers 4 years. So, the change in population is \(340 - 285 = 55\).
This slope is negative, indicating a decrease, and is calculated as:

• \( m = \frac{285 - 340}{4 - 0} = -13.75 \)

Negative slope values signify a downward trend in the population, showing us how many owls are lost each year. In this case, the population decreases by approximately 13.75 owls per year over this period.

Linear Equation

Linear equations are equations of the first order, represented in the standard format as \( y = mx + b \). Here,

• \( m \) is the slope (rate of change),
• \( x \) is the independent variable (in this case, years since 2003),
• and \( b \) is the y-intercept (initial population when \( x = 0 \)).

In the owl population problem, the linear equation models how the population changes over time.
The equation is expressed as \( P(x) = -13.75x + 340 \). This equation implies:

• When \( x = 0 \), the population is exactly 340 owls,
• The population decreases by 13.75 owls for each additional year.

Understanding this linear model allows us to predict future owl populations by simply plugging in the number of years since 2003 into the equation to find \( P(x) \).

Population Prediction

Population prediction using a linear model involves substituting a given time into the linear equation to find the projected population. For instance, predicting the population in a future year like 2012 involves identifying how many years have passed since our base year.
Let's calculate for 2012:

• 2012 is 9 years since 2003 (since \( x = 9 \)).

Substitute \( x = 9 \) into the equation \( P(x) = -13.75x + 340 \):

• \( P(9) = -13.75 \times 9 + 340 \)
• \( = -123.75 + 340 \)
• \( = 216.25 \)

Thus, the model predicts approximately 216 owls for 2012. Linear prediction models are handy for estimating how certain factors, such as population, will change over time based on past data.